Two tangent galvanometers having coils of the same radius are connected in series. A current flowing in them produces deflections of `60^(@)` and `45^(@)` respectively. The ratio of the number of turns in the coils is

Explanation

To solve the problem, we need to find the ratio of the number of turns in the coils of two tangent galvanometers connected in series, given their deflections when the same current flows through them.

### Step-by-Step Solution:

1. **Understanding the Tangent Galvanometer**:
A tangent galvanometer measures the current by the angle of deflection it produces in a magnetic field. The relationship between the current (I), the number of turns (n), and the angle of deflection (θ) is given by:
\[ I = k \cdot \tan(\theta) \]
where \( k \) is a constant that is inversely proportional to the number of turns \( n \) in the coil.

2. **Setting Up the Equations**:
For the first galvanometer with deflection \( θ_1 = 60^\circ \):
\[ I = k_1 \cdot \tan(60^\circ) \]
For the second galvanometer with deflection \( θ_2 = 45^\circ \):
\[ I = k_2 \cdot \tan(45^\circ) \]

3. **Using the Tangent Values**:
We know that:
\[ \tan(60^\circ) = \sqrt{3} \quad \text{and} \quad \tan(45^\circ) = 1 \]
Thus, we can rewrite the equations as:
\[ I = k_1 \cdot \sqrt{3} \quad \text{(1)} \]
\[ I = k_2 \cdot 1 \quad \text{(2)} \]

4. **Equating the Currents**:
Since the current \( I \) is the same in both cases, we can set the equations equal to each other:
\[ k_1 \cdot \sqrt{3} = k_2 \]

5. **Finding the Ratio of Constants**:
Rearranging gives:
\[ \frac{k_2}{k_1} = \sqrt{3} \]

6. **Relating Constants to Number of Turns**:
Since \( k \) is inversely proportional to the number of turns \( n \):
\[ k \propto \frac{1}{n} \]
Therefore, we can express the ratio of the number of turns as:
\[ \frac{n_1}{n_2} = \frac{k_2}{k_1} \]

7. **Final Ratio Calculation**:
Substituting the value we found:
\[ \frac{n_1}{n_2} = \sqrt{3} \]

### Conclusion:
The ratio of the number of turns in the coils of the two tangent galvanometers is:
\[ \frac{n_1}{n_2} = \sqrt{3} \]